Extensions of tensor categories by finite group fusion categories

• Published in: Mathematical proceedings of the Cambridge Philosophical Society Vol. 170; no. 1; pp. 161 - 189
• Main Author: NATALE, SONIA
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.01.2021
• Subjects: 18D10; 16T05
• ISSN: 0305-0041; 1469-8064
• DOI: 10.1017/S0305004119000434

We study exact sequences of finite tensor categories of the form Rep G → → , where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on such that is equivalent to a certain associated crossed extension (G,Γ) of . Dually, we show that an exact sequence of finite tensor categories VecG → → induces an Aut(G)-grading on whose neutral homogeneous component is a (Z(G), Γ)-crossed extension of a tensor subcategory of . As an application we prove that such extensions of are weakly group-theoretical fusion categories if and only if is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.